Stability Analysis Of Impulsive Functional Differential Systems With Delay-related Impulsive Function

As is well known, the stability analysis of impulsive differential systems is an important research branch of nonlinear systems of dynamics theoretical research, and is also an focus and difficult problem of nonlinear dynamic systems research in the world. Many real problems can be investigated deeply by stability analysis because of the complexity of nonlinear impulsive differential systems. So far, there are many results for stability and extension of Razumikhin technique of impulsive differential systems, see [1-35]. However, most of the results for stability and asymptotical stability always depend on the general impulsive conditions in single sense[25-35,37-43 ]. And to the best of authors' knowledge, there are very few results for stability and asymptotical stability of impulsive differential systems in which the impulsive conditions depend on time delays, see [36]. In practice, especially in the neural network optimization calculation and the design of quick search network capacity[37,38], the impulsive conditions always depend on the time delays or depend on them indirectly. Hence, the impulsive differential systems in general sense will not be able to describe the development of objective things exactly. Thus the research on stability theory of impulsive differential systems in which the impulsive conditions depend on time delays has a special guiding significance.In this context, we consider the following systems:we give some sufficient conditions for the uniform stability and uniform asymptotic stability of impulsive differential systems in which the impulsive conditions depend on time delays (â… ) by utilizing the improved Razumikhin conditions. The uniform stability and uniform asymptotic stability for addressed systems are also investigated by utilizing the Lyapunov functional partial variables methods.In Chapter one, we firstly investigated the uniform stability and uniform asymptotical stability for system (â… ) by using Lyapunov functional and Razumikhin technique. Some sufficient conditions for uniform stability and uniform asymptotical stability are obtained. Secondly, by using the Lyapunov functional partial variables methods, some other sufficient conditions for uniform asymptotical stability are derived under relatively weaker restrictions on Razumikhin conditions and stronger restrictions on impulsive conditions. Finally, some sufficient conditions for strict stability of system (â… ) are studied by using a proper Lyapunov functional.In Chapter two, we give some new results for exponential stability of system (â… ) by using Lyapunov functional and Razumikhin technique. Some illustrative examples are given to demonstrate the effectiveness of the obtained result.